Problem 43 - Algebra A Combined Function

Question

$$ \begin{aligned} &\text { Graph the system: }\\\ &\left\\{\begin{array}{l} y \leq x^{2} \\ y \geq x+2 \\ x \geq 0 \\ y \geq 0 \end{array}\right. \end{aligned} $$

Step-by-Step Solution

Answer

The feasible region is in the first quadrant, below the parabola and above the line.

Step 1: Graph the Inequality \(y \leq x^2\)

Start by graphing the equation \(y = x^2\), which is a standard upward-opening parabola. Since the inequality is \(y \leq x^2\), shade the region that is below or on the parabola.

Step 2: Graph the Inequality \(y \geq x + 2\)

Graph the line \(y = x + 2\), which is a straight line with a slope of 1 and a y-intercept at 2. For the inequality \(y \geq x + 2\), shade the region that is above or on this line.

Step 3: Graph the Inequality \(x \geq 0\)

The inequality \(x \geq 0\) represents all the points on or to the right of the y-axis. Shade this entire region including the y-axis.

Step 4: Graph the Inequality \(y \geq 0\)

This inequality means that only the points on or above the x-axis are valid. Shade the entire region above or on the x-axis.

Step 5: Find the Feasible Region

The feasible region for this system is where all the shaded regions overlap. It should only include points where all four conditions are satisfied: below the parabola \(y = x^2\), above or on the line \(y = x + 2\), to the right of the y-axis \(x = 0\), and above the x-axis \(y = 0\).

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values, indicating that one is either less than, greater than, or equal to another. In graphing systems of inequalities, we often encounter symbols like \(<, >, \leq, \geq\). Each of these symbols tells us how to shade the region on a graph.

For instance, if you have an inequality such as \(y \leq x^2\), it means that you're looking at all the points on a graph where the y-values are less than or equal to the value given by the equation \(y = x^2\). This inequality describes a parabola, and the shaded area will be below the curve. Inequalities help specify ranges of solutions rather than just a single solution.

When graphing inequalities, it is crucial to pay attention to whether the inequality is strict (using \(<\) or \(>\)) or inclusive (using \(\leq\) or \(\geq\)).

• Strict inequalities use dashed lines because the solutions are not included on the boundary.
• Inclusive inequalities—like some in our system—use solid lines to show that solutions on the line are valid.

Parabolas

A parabola is a U-shaped curve that you’ll often see in quadratic functions, typically represented as \(y = x^2\). In this context, it arises from inequalities that include quadratic terms. Here, the inequality is \(y \leq x^2\), which conveys the area below or on the parabola is included in the solution set.

To graph a parabola properly, it's essential to understand its basic characteristics:

• The vertex of the parabola is a critical point that serves as the starting reference. For \(y = x^2\), the vertex is at the origin (0, 0).
• The parabola opens upwards if the term \(x^2\) is positive and downwards if negative.

When dealing with parabolas and inequalities, shading accurately is vital. For \(y \leq x^2\), you shade the region underneath the parabola because those are the y-values that satisfy the "less than or equal to" condition.

Linear Equations

Linear equations are foundational in understanding systems of inequalities. They appear as straight lines on a coordinate plane. The general form of a linear equation is \(y = mx + b\), where \(m\) represents the slope and \(b\) the y-intercept.

In the given exercise, one of the linear equations is \(y \geq x + 2\). This indicates a line with a slope of 1 that intercepts the y-axis at 2. The inequality part, \(\geq\), tells us to shade the region above the line.

• To plot a linear equation, identify a minimum of two points using the slope and y-intercept, draw a straight line through these points.
• If the inequality is inclusive, like \(y \geq x + 2\), a solid line is used as solutions include points on the line.

Understanding how the slope affects the direction and steepness of the line provides insights into how it interacts with other equations in the system, defining the feasible region.

Coordinate Plane

A coordinate plane, also known as the Cartesian plane, is a two-dimensional surface where points are defined by a pair of numerical coordinates. These coordinates are written as \((x, y)\). The plane is divided by a horizontal x-axis and a vertical y-axis, meeting at the origin—point \((0,0)\).

Understanding the layout of the coordinate plane helps when plotting inequalities, as it provides the foundation for identifying the regions each inequality represents.

• The x-axis divides the plane into upper and lower halves, corresponding to positive and negative y-values.
• The y-axis separates it into left and right halves for negative and positive x-values.

For graphing systems of inequalities, the coordinate plane becomes a visual tool to illustrate where solutions to an inequality are valid. As students shade regions based on inequalities like \(x \geq 0\) and \(y \geq 0\), they utilize half-planes determined by the axes, fully understanding how inequalities shape their feasible region.